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Permutations Related to Secant, Tangent and Eulerian Numbers

Published online by Cambridge University Press:  20 November 2018

Morton Abramson
Morton Abramson*
Affiliation:
Department of Mathematics, York University, 4700 Kelle Street, Downsview, Ont., M3J IP3
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It is well known that

1

where An denotes the number of "up-down" or alternating permutations

2

of 1, 2, …, n. The numbers A2n and A2n+1 are known as the secant and tangent numbers respectively and A2n = (—l)"E2n, where En is the Euler number.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 3 , 01 September 1979 , pp. 281 - 291
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Abramson, Morton, Some generalized Eulerian numbers, Proc. Fifth Southeastern Conference on Combinatorics, graph theory and computing, Florida Atlantic University, Boca Raton, Florida, 1974, pp. 159-172.Google Scholar
2. Abramson, Morton, A simple solution of Simon Newcomb's problem, J. of Comb. Theory, “A”, 18 (1975), pp. 223-225.Google Scholar
3. Abramson, Morton, A note on permutations with fixed pattern, J. of Comb. Theory, “A”, 19 (1975), pp. 237-239.Google Scholar
4. Abramson, Morton, Enumeration of sequences by levels and rises, Discrete Math., 12 (1975), pp. 101-112.Google Scholar
5. Abramson, Morton, Sequences by number of w-rises, Canad. Math. Bull., 18 (1975), pp. 317-319.Google Scholar
6. Abramson, Morton and Promislow, David, Enumeration of arrays by column rises, J. of Comb. Theory, “A”, 24 (1978), pp. 247-250.Google Scholar
7. André, D., Développements de see x et de tan x, C. R. Acad. Sci. Paris, 88 (1879), pp. 965-967.Google Scholar
8. André, D., Sur les permutations alternées, J. Math. Pures Appl., 7 (1881), p. 167.Google Scholar
9. Blundon, W. J., Solution to “A permutation problem”, 4755 [1957, 596], Amer. Math. Monthly, 65 (1958), p. 533.Google Scholar
10. Carlitz, L., Some arithmetic properties of the Oliver functions, Math. Ann., 128 (1955), pp. 412-419.Google Scholar
11. Carlitz, L., Eulerian numbers and polynomials, Math. Magazine, 32 (1959), pp. 247-260.Google Scholar
12. Carlitz, L., Eulerian numbers and polynomials of higher order, Duke Math. J., 27 (1960), pp. 401-423.Google Scholar
13. Carlitz, L., A note on Eulerian numbers, Arch. Math., 14 (1963), pp. 383-390.Google Scholar
14. Carlitz, L., Extended Bernoulli and Eulerian numbers, Duke Math. J., 31 (1964), pp. 667-689.Google Scholar
15. Carlitz, L., Enumeration of sequences by rises and falls: a refinement of the Simon Newcomb problem, Duke Math. J., 39 (1972), pp. 267-280.Google Scholar
16. Carlitz, L., Eulerian numbers and operators, The Theory of Arithmetic Functions, pp. 65-70, Lecture Notes in Math., Vol. 251, Springer, Berlin, 1972.Google Scholar
17. Carlitz, Leonard, Enumeration of a special class of permutations by rises, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 412-460 (1973), pp. 189-196.Google Scholar
18. Carlitz, L., Enumeration of permutations by rises and cycle structure, J. Reine Angew. Math., 262/263 (1973), pp. 220-233.Google Scholar
19. Carlitz, L., Enumeration of up-down permutations by number of rises, Pacifie J. Math., 45 (1973), pp. 49-58.Google Scholar
20. Carlitz, L., Enumeration of up-down sequences, discrete Math., 4 (1973), pp. 273-286.Google Scholar
21. Carlitz, L., Eulerian numbers and operators, collect. Math., 24 (1973) pp. 175-200.Google Scholar
22. Carlitz, L., Permutations with prescribed pattern, Math. Nachr., 58 (1973), 31-53.Google Scholar
23. Carlitz, L., Permutations and sequences, Advances in Math., 14 (1974), pp. 92-120.Google Scholar
24. Carlitz, L., A combinatorial property of q-Eulerian numbers, Amer. Math. Monthly, 82 (1975), pp. 51-54.Google Scholar
25. Carlitz, L., Generating functions for a special class of permutations, Proc. Amer. Math. Soc, 47 (1975), pp. 251-256.Google Scholar
26. Carlitz, L., Permutations, sequences, and special functions, SLAM Rev., 17 (1975), pp. 298-321.Google Scholar
27. Carlitz, L., Combinatorial property of a special polynomial sequence, Canad. Math. Bull., 20 (1977), pp. 183-188.Google Scholar
28. Carlitz, L., Kurtz, D. C., Scoville, R., and Stackelberg, O. P., Asymptotic properties of Eulerian numbers, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 23 (1972), pp. 47-54.Google Scholar
29. Carlitz, L. and Riordan, J., Congruences for Eulerian Numbers, Duke Math. J., 20 (1953), pp. 339-343.Google Scholar
30. Carlitz, L., Roselle, D. P. and Scoville, R. A., Permutations and sequences with repetitions by number of increases, J. Comb. Theory, 1 (1966), pp. 350-374.Google Scholar
31. Carlitz, L. and Scoville, Richard, Tangent numbers and operators, Duke Math. J., 39 (1972), pp. 413-429.Google Scholar
32. Carlitz, L. and Scoville, Richard, Up-down sequences, Duke Math. J., 39 (1972), pp. 583-598.Google Scholar
33. Carlitz, L. and Scoville, Richard, Enumeration of rises and falls by position, Discrete Math., 5, (1973), pp. 45-59.Google Scholar
34. Carlitz, L. and Scoville, Richard, Enumeration of permutations by rises, falls, rising maxima, and falling maxima, Acta Math. Acad. Sci. Hungar., 25 (1974), pp. 269-277.Google Scholar
35. Carlitz, L. and Scoville, Richard, Generalized Eulerian numbers: combinatorial applications, J. Reine Angew. Math., 265 (1974), pp. 110-137.Google Scholar
36. Carlitz, L. and Scoville, Richard, Enumeration of up-down permutations by upper records, Monatshefte fur Mathematik, 79 (1975), pp. 3-12.Google Scholar
37. Carlitz, L. and Scoville, Richard, Generating functions for certain types of permutations, J. Combinatorial Theory “A”, 18 (1975), pp. 262-275.Google Scholar
38. Carlitz, L. and Scoville, Richard, Eulerian numbers and operators, Fibonacci Quart., 13 (1975), pp. 71-83.Google Scholar
39. Carlitz, L. and Scoville, Richard, Some permutation problems, J. Comb. Theory “A, 22 (1977), pp. 129-145.Google Scholar
40. Carlitz, L., Scoville, Richard and Vaughan, Theresa, Enumeration of permutations and sequences with restrictions, Duke Math. J., 40 (1973), pp. 723-741.Google Scholar
41. Carlitz, L., Scoville, R. and Vaughan, T., Enumeration of pairs of permutations and sequences, Bull. Amer. Soc, 80 (1974), pp. 881-884.Google Scholar
42. Carlitz, L., Scoville, R., and Vaughan, T., Enumeration of pairs of permutations, Discrete Math., 14 (1976), pp. 215-239.Google Scholar
43. Carlitz, L., Scoville, R. and Vaughan, T., Enumeration of pairs of sequences by rises, falls, and levels, Manuscripta Math., 19 (1976), pp. 211-243.Google Scholar
44. Carlitz, L. and Vaughan, T., Enumeration of sequences of given specification according to rises, falls, and maxima, Discrete Math., 8 (1974), pp. 147-167.Google Scholar
45. Comtet, L., Analyse Combinatoire, vol. 1, pp. 63-64, vol. 2, pp. 82-86, Paris, 1970.Google Scholar
46. Debruijn, N. E., Permutations with given ups and downs, Nieuw Archief voor Wiskunde, 18 (1970), pp. 61-65.Google Scholar
47. Dillon, J. F. and Roselle, D. P., Eulerian numbers of higher order, Duke Math. J., 35 (1968), pp. 247-256.Google Scholar
48. Dillon, J. F. and Roselle, D. P., Simon Newcomb's problem, SIAM J. Appl. Math., 17 (1969), pp. 1086-1097.Google Scholar
49. Donaghey, Robert, Alternating permutations and binary increasing trees, J. Comb. Theory, “A”, 18 (1975), pp. 141-148.Google Scholar
50. Drane, F. B. and Roselle, D. P., A sequence of polynomials related to the Eulerian polynomials, Utilitas Math., 9 (1976), pp. 33-37.Google Scholar
51. Dumont, Dominique, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), pp. 305-318.Google Scholar
52. Entringer, R. C., A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Arch. V. Wiskunde, 14 (1966), pp. 241-246.Google Scholar
53. Etringer, R. C., A note on enumeration of permutations of ( 1, …, n) by number of maxima, Gac. Mat. (Madrid), 23 (1971), pp. 67-69.Google Scholar
54. Euler, L., Institutiones Calculi Differentiate, St. Petersburg, 1755.Google Scholar
55. Foata, Dominique, Groupes de réarrangements et nombres d' Euler, C. R. Acad. Sci. Paris Sér. A-B, 275 (1972), A 1147-A 1150.Google Scholar
56. Foata, Dominque and Schùtzenberger, Marcel-P., Théorie géométrique des polynômes eulériens, in “Lecture Notes in Mathematics,” No. 138, Springer-Verlag, New York, 1970.Google Scholar
57. Foata, D. and Schùtzenberger, M. P., Nombres d' Euler et permutations alternantes, A Survey of Combinatorial Theory, pp. 173-187, North-Holland, Amsterdam, 1973.Google Scholar
58. Foata, Dominque and Strehl, Volker, Euler numbers and variations of permutations, Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo I, pp. 119-131. Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Roma, 1976.Google Scholar
59. Foulkes, H. O., Enumeration of permutations with prescribed up-down and inversion sequences, Discrete Math., 15 (1976), pp. 235-252.Google Scholar
60. Foulkes, H. O., A nonrecursive combinatorial rule for Eulerian numbers, J. Comb. Theory, “A”, 22 (1977), pp. 246-248.Google Scholar
61. Gessel, Ira Martin, Generating Functions and Enumeration of sequences, Ph.d. Thesis, M.I.T. 1977.Google Scholar
62. Gessel, Ira and Stanley, Richard P., Stirling Polynomials, J. of Comb. Theory, 24 (1978), pp. 24-33.Google Scholar
63. Gould, H. W., Comment on Problem 67-5, “up-down” permutations, SIAM Review, 10 (1968), pp. 225-226.Google Scholar
64. Gould, H. W., Explicit formulas for Bernoulli numbers, Amer. Math. Monthly, 79 (1972), pp. 44-51.Google Scholar
65. Johnson, Allan Wm. Jr. Solution to, “ A difference equation in two variables,” Problem E 2609 [1976, 567], Amer. Math. Monthly, 84 (1977), pp. 826-827.Google Scholar
66. Knop, Robert E., A note on hypercube partitions, J. Comb. Theory, “A”, 15 (1973), pp. 338-342.Google Scholar
67. Kreweras, Germain, Sur une extension du problème dit “de Simon Newcomb”, C.R. Acad. Sci. Paris Sér. A-B, 263 (1966), pp. A43-A45.Google Scholar
68. Kurtz, David C., A note on concavity properties of triangular arrays of numbers, J. of Comb. Theory “A”, 13 (1972), pp. 135-139.Google Scholar
69. Netto, E., Lehrbuch der Combinatorik, Berlin, 1927.Google Scholar
70. Olivier, L., Bemerkungen uber eine Art von Funktionen, welche Eigenschaften haben, wie die Cosinus und Sinus, J. Reine Angew. Math., 2 (1827), pp. 243-251.Google Scholar
71. Riordan, J., Triangular permutation numbers, Proc. Amer. Math. Soc, 2 (1951), pp. 429- 432.Google Scholar
72. Riordan, J., An Introduction to Combinatorial Analysis, J. Wiley, New York, 1958.Google Scholar
73. Roselle, D. P., Permutations by number of rises and successions, Proc. Amer. Math. Soc, 19 (1968), pp. 8-16.Google Scholar
74. Rosen, J., The number of product-weighted lead codes for ballots and its relation to the Ursell-functions of the Linear Ising Model, J. Comb. Theory, “A”, 20 (1976), p. 377.Google Scholar
75. Sorin, B., La distribution combinatoire du nombre de différences premières positives, Rev. Inst. Internat. Statist., 39 (1971), pp. 9-20.Google Scholar
76. Stanley, Richard P., Ordered Structures and Partitions, Memoirs of the Amer. Math. Soc, No. 119, Amer. Math. Soc, Providence, R. L, 1972.Google Scholar
77. Stanley, Richard P., Binomial posets, Môbius inversion, and permutation enumeration, J. Combinatorial Theory “A”, 20 (1976), pp. 336-356.Google Scholar
78. Strehl, Volker, Enumeration of alternating permutations according to peak sets, J. of Comb. Theory, “A”, 24 (1978), pp. 238-240.Google Scholar
79. Tanny, S., A probabilistic interpretation of Eulerian numbers, Duke Math. J., 40 (1973), pp. 717-722.Google Scholar
80. West, Don, Solution to “The Smith college diploma problem” Problem E 2404 [1973, 316], Amer. Math. Monthly, 81 (1974), pp. 286-289.Google Scholar
81. Worpitzky, J., Studien über die Bernoullischen und Eulerschen Zahlen, J. fur die reine und angewandte Math., 94 (1883), pp. 203-232.Google Scholar